A species (or taxon) response curve (SRC) defines the probability of an observed count of a species, or taxon, as a function of the environmental variable of interest. An SRC is defined by five (initially unknown) parameters. These parameters specify the environmental optimum and tolerance of each taxon, together with the probability of its presence, and its expected abundance under optimal conditions (see Sect. 2.3 for details).

BUMPER considers 2560 possible SRCs for each taxon

The model of Holden et al. (2008) considered 8000 SRCs. We here reduce the SRC resolution from (20, 4, 5, 5, 4) to (10, 4, 4, 4, 4) in order to benefit from the ∼ 3-fold increase in computational speed.

To reconstruct the environment from an observed fossil count yk0 of each taxon k within the fossil assemblage, the probability-weighted SRCs are used to derive likelihood functions for each taxon of the reconstructed variable x. Here we again use the Bayes relationship, this time stating that the probability that a reconstructed value is correct in the light of an observed taxon count is proportional to the probability that the taxon count would be observed in that environment. Considering for now a single observed taxon, Bayes' equation can be written as follows: probx|yk0∝probyk0|x×prob(x).

We derive a normalized likelihood function for the taxon, considering only the first term on the right hand side of Eq. (3). This allows us to treat the function as a probability distribution of the environmental variable, given no prior knowledge and a count of this single taxon in isolation. As we do not know the true SRC of the taxon with certainty (in general, the calibration will have resulted in many SRCs with non-zero probability), the likelihood function is derived from all significant SRCs, combined using the probability weights calculated in Sect. 2.1, applying the law of total probability: Lyx|yk0∝probyk0|x=∑jprobyk0,SRCjk|x=∑jprobSRCjk×probyk0|SRCjk,x. This expression is evaluated at 100 evenly spaced points across a range that comfortably spans the training-set environmental range (see Sect. 2.4), and is normalized to 1. (We note that this normalization step is required to build the blended likelihood function in Eq. 6.)

An alternative and less-constrained presence–absence likelihood function can be derived (and normalized as Eq. 4): Lpx|yk0∝probyk0>0|x=∑jprobSRCjk×probyk0>0|SRCjk,x. BUMPER derives a linear combination of these two functions, as follows: Lx|yk0=1-ηLyx|yk0+ηLpx|yk0. The motivation for this blended likelihood function is that Ly provides a tighter constraint on the posterior, but the wide tails contributed by Lp reduce the possibility of the correct solution being ruled out by an outlying taxon, for instance resulting from misidentification or unusual taphonomic processes. We assume η=0.5, except when building a purely binary, presence–absence model, when η=1.0. Hereafter, we refer to L simply as the likelihood function of a taxon.

The posterior probability distribution for the reconstructed variable probx is derived by combining any prior knowledge probx′ with the product of likelihood functions of all considered taxa in the assemblage, as follows: probx∝probx′×∏kLx|yk0. The reconstruction is evaluated at the same 100 points as the likelihood function, and the probabilities are normalized to 1.

A point estimate for the reconstructed variable is derived as follows: x^=∫xprobxdx. A simple uncertainty metric for the reconstruction is derived as follows: Δ=∫x-x^2probxdx.

To apply Bayes' equation we require an expression for the probability of observing some count y as a function of the environmental variable x. The assumption of a unimodal response leads us to assume that the expected abundance (given presence) Nik of taxon k in site i can be fitted by a Gaussian response curve (ter Braak and Barendregt, 1986) about some optimum value uk of the environmental variable xi, written as follows: Nik=Nke-xi-uk2-xi-uk22tk22tk2. Here Nk is the expected abundance (given presence) at the taxon optimum, and the tolerance tk is a measure of how rapidly the expected abundance falls off away from this optimum.

The probability of presence (e.g. used to calculate the Lp likelihood function, Eq. 5) is also assumed to follow a Gaussian distribution about the same optimum, though not necessarily of the same tolerance, and can be written in terms of the expected abundance, as follows: pik=pkNikNikNkNkPk. where Pk is needed because we do not require the same tolerance for Nk and pk , although we can achieve this with Pk=1. We illustrate the role of Pk through example; Pk≪1 describes a taxon that is found at high abundance only in an environment that is relatively close to its environmental optimum, but may also be present (albeit only at low abundance) far away from this optimum.

Taxon-count distributions are represented with a hurdle model (a zero-inflated distribution with truncation at zero of the distribution of percentage counts). The probability of a non-zero count yik (used to calculate the Ly likelihood function, Eq. 4) is assumed to follow an exponential decay, with the decay constant defined by the expected count =11NikNik, normalized so that the total probability of all (non-zero) percentage counts equals the probability of presence pik, and is expressed as a continuous distribution, truncated at 0 and 100: probyik|xi=pikpikNikNikexp⁡-yik-yikNikNik1-exp⁡-100-100NikNik. The denominator normalizes the integral over the range 0<yik≤100. We note that the denominator is missing from the description of Holden et al. (2008), although it was correctly implemented in the model itself. The probability of absence (yik=0) is given by 1-pik.

The SRCs are defined by five parameters: species or taxon optimum uk, tolerance tk, tolerance scaling Pk, probability of presence at environment optimum pk, and expected abundance (given presence) at environmental optimum Nk. In general we have little a priori knowledge of appropriate SRC values, and require their priors to be uninformative. However, for computational tractability, we need to eliminate implausible parameter values.

Previous applications (Holden et al., 2008; Matthews-Bird et al., 2016) used uniform priors for the SRC parameters over ranges that were chosen largely on the basis of subjective judgement (and ascribing a probability of zero outside of those ranges). However, requiring subjective judgement is not desirable given our aim to maximize the user-friendliness of the model, and it can be rather time consuming. To prevent subjectivity, BUMPER introduces a simple automated process, based upon an indicative taxon tolerance t′ that is a characteristic of the training set: t′=1m∑k1nk∑ixi-ukWA2. For each taxon k we consider the environmental variable xi at each site where the taxon is present, and derive a root mean square (RMS) distance from the weighted-average optima, ukWA. We average across the m taxa that have at least two training-set observations (nk≥2). The weighted-average optima are defined in the usual way: ukWA=∑iyikxi∑iyikxi∑iyik∑iyik. This metric t′ is used to automate the definition of plausible ranges for taxon optima and tolerance. The approach we take is analogous to precalibrating the parameters of a physical model (Edwards et al., 2011). We only attempt to rule out implausible parameter values and we ascribe equal probabilities to all plausible parameter combinations. The precalibration only seeks to eliminate SRCs with a probability close to zero, and is thereby designed to avoid the trap of over-constraining the posterior by using the training-set data twice.

We consider 10 possible values for the taxon optima, and 4 possible values for each of the 4 other parameters, giving a total of 2560 SRC combinations. The priors for the five SRC parameters are uniform within specified ranges:

Optima uk are allowed to take values in the range xmin⁡-t′ to xmax⁡+t′, where xmax⁡ and xmin⁡ are the extremes of the training-set environments. We allow optima beyond the sampled environment range, but optima beyond this extended range are considered unlikely. These ranges are approximately consistent with the subjective priors used in Holden et al. (2008), being training set ±0.5 pH units (t′=0.63 pH units), and Matthews-Bird et al. (2016), being training set ±5 ∘C (t′=3.0 ∘C).

While aspects of the motivation for ranges 3–5 are discussed above, other aspects are not clearly defined by logical considerations, these being the upper limits for pk and Nk, and the lower limit for Pk. These ranges were selected empirically on the basis of application of the model to three training sets, being chironomid-based temperature (Matthews-Bird et al., 2016), diatom-based pH (Stevenson et al., 1991) and pollen-based temperature (Bush et al., 2017). We define the posterior value for a parameter to be the probability-weighted value of that parameter across the 2560 SRCs of the respective taxon. The posterior values of pk, Nk, and Pk for each taxon were sorted from smallest to largest and are plotted sequentially in Fig. 1, for each of the three training sets. The horizontal grid lines represent the four values that each parameter is allowed to take. (Note that although the SRC parameters can only take one of four discrete values, the probability-weighted average can take any value within the prior range). It is apparent that the full range of allowed values is required to describe the responses of all taxa in all three training sets (i.e. because the extremes of the probability-weighted averages approach the extremes of the allowed parameter range). Furthermore it is apparent that a wider range is not required, although there is some suggestion that a slightly higher upper limit for pk could have been allowed for the SWAP data (evidenced by the fact that 7 of the 225 species take precisely the upper limit pk=2.5pk′).

We note that the indicative tolerance is also used to define the environmental range considered in the reconstruction (Sect. 2.2), from xmin⁡-6t′toxmax⁡+6t′. Significant probabilities beyond this range are unlikely given the constraints imposed upon the optima and tolerances. In any event, as with any transfer function, the model should not be applied under suspected extrapolation far beyond the training-set environment.

We generate an artificial training set for an arbitrary environmental variable, under the following assumptions. i.

The training-set sites have an observed environment that is randomly sampled from a uniform distribution over some range EL to EH. This choice is arbitrary; we select EL=100, EH=200. The SRC parameters and performance statistics that follow are expressed as a percentage of the environmental gradient EH-EL=100.

The values of the abundance parameter Nk are drawn from a modified exponential distribution (cf. Eq. 15) with scale parameter βN. The value of βN is not an input because an arbitrary choice will lead to an expected total number of counts (i.e. the sum of all taxon counts within a site) that differs from 100 %. An appropriate value of βN is derived from a Monte Carlo approach; the expected total count is derived from an exploratory 10 000-site training set with βN=10 % and then βN is scaled appropriately and the scaled value applied to generate the actual training set. BUMPER does not model a multinomial probability distribution so that percentage counts generated under this approach are not constrained to sum precisely to 100 %, even when the appropriate value of βN is chosen. A pragmatic approach is taken to address this issue by randomly drawing assemblages, accepting only those with a total count of between 90 and 110 % until the required number of sites has been generated. In accepted sites the counts are scaled to total 100 %.

In this section, we cross-validate transfer functions derived from artificial training sets with different characteristics to investigate how these characteristics affect the reconstruction performance. The characteristics we consider are assemblage richness and assemblage tolerance. We define the characteristic assemblage richness as the average number of taxa found in the training-set sites. This is varied in the artificial ensemble through the exponential scale parameter βp. A low value of βp ascribes low probabilities of presence, even under optimal conditions, and leads to taxon-poor assemblages. We define the characteristic assemblage tolerance to be the indicative tolerance t′ (Sect. 2.4). This is adjusted through the prior range for tolerance, tL to tH. For each assemblage type considered, training sets are generated from 25, 100, 250, and 1000 sites, and are comprised of the same 100 randomly generated artificial taxa.

We consider three assemblage types.

Low richness, high tolerance; βp=10 %, tL=15 %, tH=25 %, (solved βN=40 %). The assemblage richness that results from these assumptions is 5.7, i.e. the training-set sites contain on average 5.7 taxa. Tolerances are typically 20 % of the environmental gradient.

The chironomid training set of Matthews-Bird et al. (2016) comprises 59 lakes across tropical South America. The prediction uncertainty associated with this dataset (WAPLS RMSEP 2.4 ∘C) is approximately twice the uncertainty that has been achieved for European chironomid transfer functions, which typically approach ∼ 1 ∘C (e.g. Brooks et al., 2012). It has been postulated that this reduced performance is caused by the small size of the training set and uneven sampling across the environmental gradient (Matthews-Bird et al., 2016).

The default model (yk> 2 %, η=0.5) exhibits an RMSEP of 2.41 ∘C (Fig. 3a), and 97 % of cross-validated reconstructions lie within ±2Δ of the observed temperature. However, the best model (RMSEP 2.27 ∘C) is the presence–absence model that requires 2 % abundance to include a species in a reconstruction (Fig. 3b). This model version is the most conservative and might therefore be expected to be the least well performing. However, the training set is small, so the SRCs are not expected to be well defined, and furthermore the number of counts per site is relatively low. These data may therefore favour a more conservative model that does not over-constrain the reconstructions.

When all taxa are included, the reconstruction uncertainty is reduced as expected, from 12.0 to 11.0 % in the default model and from 13.3 to 11.8 % in the presence–absence model. However, this is not accompanied by a reduction in RMSEP. The RMSEP increases slightly from 9.7 to 9.8 % in the default model and from 9.1 to 9.6 % in the presence–absence model. These data support our choice to retain the 2 % abundance threshold in the default model. Including very low abundance counts increases the computational demand, does not in general improve model performance, and is likely to understate the uncertainty associated with the reconstruction. We note for completeness that the default BUMPER model was also applied in Matthews-Bird et al. (2016) and has a RMSEP of 2.37 ∘C. The slight difference in performance here results from the different SRC priors used (see Sect. 2.4).

The Matthews-Bird training set can be broadly classified as being low taxon richness (7.0), narrow tolerance (12 %), and relatively poorly sampled (sampling density 7.0). We generated an ensemble of artificial datasets with approximately those characteristics (59 lakes, 59 taxa, βp=40 %, tL=8 %, tH= 18 %, yielding an ensemble-averaged species richness of 6.7). The WAPLS1 transfer functions built from these data exhibits a mean RMSEP of 6.3 ± 0.6 % of the environmental gradient. Although a direct comparison is difficult, given that an improvement in performance is expected under idealized assumptions, these statistics are broadly consistent with those of the real transfer functions (ranging from 9.1 to 9.8 %).

We consider the training-set characteristics in order to investigate whether they might explain the larger uncertainties of the Matthews-Bird transfer function when compared with European chironomid transfer functions. First, we increase the idealized-analogue training-set size from 59 lakes to 118 lakes, thereby better defining the taxon characteristics. This improves the WAPLS1 RMSEP from 6.3 ± 0.6 % to 5.9 ± 0.4 %. Second, we increase the taxon richness from 6.7 to 13.5 by doubling the number of species to 118. This improves the WAPLS1 performance to 5.1 ± 0.5 %. Although both factors are likely to contribute to explaining the reduced uncertainty of the European transfer functions, in isolation they do not appear sufficient. We consider a specific example. The Norwegian-chironomid training set of Brooks et al. (2012) comprises 140 taxa sampled from 157 lakes, and was used to construct a WAPLS2 transfer function for July temperature with RMSEP of 1.06 ∘C when four outliers are deleted (Brooks et al., 2012). We note that the Matthews-Bird training set reconstructs the mean annual temperature, not the July temperature. The Brooks and Birks training set has an average richness of 24 and an indicative tolerance of 16 %. An idealized dataset was built with these characteristics (159 lakes, 140 taxa, βp=55 %, tL=11 %, tH=21 %, yielding an ensemble-averaged taxon richness of 22) and was found to exhibit a WAPLS1 RMSEP of 5.1 ± 0.3 % (= 0.64 ∘C). When expressed as a percentage of the environmental gradient, modest improvements in performance are apparent, consistent with that expected from increased training-set size and species richness. However, most of the improvement in predictive power is due to tolerance, but expressed in absolute terms. The Matthews-Bird indicative tolerance is 3.2 ∘C. The Brooks and Birks indicative tolerance is 2.1 ∘C. These factors together (increased training-set size, increased taxon richness and, most significantly, narrower absolute tolerances) appear sufficient to explain the improved performances of the Brooks and Birks transfer function over the Matthews-Bird transfer function.

The SWAP training set (Stevenson et al., 1991) was developed as part of a substantial scientific effort directed at understanding the impacts of acid rain. Taxonomic workshops resolved problems of nomenclature, splitting and amalgamation of species, and identification criteria (Munro et al., 1990). Approximately 500 counts per sample were made. The training set was statistically pruned (Birks et al., 1990) to leave a high-quality dataset of 267 taxa in 167 sites. The best WAPLS component has RMSEP of 0.310 pH units (Holden et al., 2008).

The best-performing BUMPER model for SWAP is the model that includes all taxa, with no threshold abundance. This model has an RMSEP = 0.321 pH units. The improvement in performance relative to the default BUMPER (yk> 2 %) threshold (RMSEP = 0.369 pH units, Fig. 3c) may reflect the high quality of the SWAP training set, so that even very low counts are unlikely to be erroneous, and therefore add value to the reconstruction. However, a weakness of the yk> 0 % model is that the posterior widths substantially understate the uncertainty; the low-count taxa narrow the posterior widths by more than is justified, given the modest improvement in performance. Using the default model (yk> 2 %, η=0.5), 92 % of cross-validated reconstructions lie within ±2Δ of the measured pH. We note that the performance of the subjectively tuned model in Holden et al. (2008) was RMSEP = 0.328 pH units.

The SWAP training set can be broadly classified as having high taxon richness (50) and broad tolerance (22 %) and being well sampled (sampling density 37). It is interesting that the RMSEP of this training set is similar to the Matthews-Bird set, both exhibiting an RMSEP of approximately 10 % of the environmental gradient. The improvements due to the high quality, taxon richness, and sampling density of SWAP are offset by the loss of precision that is achievable due to broader tolerances.

We generate an artificial training set to mimic the characteristics of SWAP (167 lakes, 225 taxa, βp=40 %, tL=17 %, tH=27 %, yielding a characteristic taxon richness of 48). The performance statistics of a model derived from these data 5.1 ± 0.3 % are again broadly consistent with, though significantly better than, the model derived from the real data (11.0 to 12.9 %). The difference between real and artificial data is greater than in the chironomid comparison. Diatoms have been applied as indicators of a wide range of environmental variables including pH, climate, water chemistry, and eutrophication (Smol and Stoermer, 2015), and it may be that the assumption of a single environmental driver of diatom-assemblage variability is less well satisfied than it is with chironomids.

The NIMBIOS dataset (Bush et al., 2017) comprises 682 samples that range from soil samples to mud–water interface samples from lakes. The dataset is of variable quality as it represents a ∼ 30-year data acquisition effort, in which time there have been substantial improvements in the ability to accurately identify pollen because new pollen keys and descriptions have become available (e.g. Roubik and Moreno, 1991; Bush and Weng, 2007).

The best BUMPER model for NIMBIOS is the full model including all taxa (η=0.5, yk>0 %), exhibiting an RMSEP = 2.51 ∘C. This represents a significant improvement to the best WAPLS model (2.92 ∘C, WAPLS component 2). As with SWAP, however, a weakness of this model is that the posteriors substantially understate the uncertainty. The default model also performs well (RMSEP = 2.88 ∘C) and exhibits posteriors that accurately reflect the uncertainty (Fig. 3e), with 93 % of cross-validated reconstructions lying within ±2Δ of the observed temperature.

The NIMBIOS training set can be classified as having high taxon richness (27) and low tolerance (12 %) and being well sampled (sampling density 33). The training set benefits from optimal characteristics in all three respects. We generate an artificial analogue of NIMBIOS (682 samples, 553 taxa, βp=15 %, tL=7 %, tH=17 %, yielding a taxon richness of 30). The BUMPER model built from this analogue exhibits an RMSEP of 3.8 ± 0.1 %. This is significantly better than the artificial analogues of the SWAP and Matthews-Bird sets, as would be expected given the optimal characteristics of the training set. However, the performance statistics in this case are substantially better than those of the model built from the real data (RMSEP = 10.1 to 12.2 %); the transfer function built from real data does not demonstrate the high performance that might be expected. The discrepancy between the idealized and real data appears to suggest that the quality of the NIMBIOS data is poorer than the other training sets. A major reason for the apparent inaccuracy may be that, unlike the other datasets, pollen is often not recognized to species level

We note that chironomids are identified to species morphotype level, or genus in some cases.